A New Wavelet-Based Mode Decomposition for …...A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition Adrien DELIÈGE

A New Wavelet-Based Mode Decomposition forOscillating Signals and Comparison with the

Empirical Mode Decomposition

Adrien DELIÈGE

University of Liège, Belgium

ITNG 2016 - 13th International Conference on Information Technology: New GenerationsLas Vegas, Nevada – April 2016

Joint work with S. NICOLAY

[email protected]

Introduction

Decomposing time series into several modes has become more and morepopular and useful in signal analysis.

Methods such as EMD or SSA (among others) have been successfullyapplied in medicine, finance, climatology, ...

Old but gold: Fourier transform allows to decompose a signal as

f (t)≈K

∑k=1

ck cos(ωk t +φk).

Problem: often too many components in the decomposition.

Idea: Considering the amplitudes and frequencies as functions of t todecrease the number of terms.

Development of an empirical wavelet mode decomposition (EWMD)method which can be used in numerous ways.

Edit: EWMD becomes WIME (wavelet-induced mode extraction).Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 Period detection skills

4 Real-life data

5 Recent improvements

6 Conclusion

Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016

Methods: EMD and WIME

Table of contents




4 Real-life data


6 Conclusion



Empirical Mode Decomposition (EMD) in a nutshell

1. Given a signal f , compute m the mean of its upper and lower envelopes.

2. Compute h = f −m and repeat steps 1. and 2. with h instead of f until h is“stable enough” and becomes an “intrinsic mode component” (IMF), f1.

3. Repeat the whole process with f − f1 instead of f .

4. The procedure always stops and the IMFs successively extractedreconstruct f accurately.

Main drawbacks include: lack of solid mathematical theoretical background,high sensitivity to noise, mode-mixing problems.

Improvements made are computationally expensive, not intuitive.



Empirical Wavelet Mode Decomposition (EWMD) – WIME

The wavelet used in this study is the function

ψ(t) =eiΩt

2√

2πe−

(2Ωt+π)2

8Ω2

(

eπtΩ +1

)

with Ω= π√

2/ ln2, which is similar to the Morlet wavelet. The Fouriertransform of ψ is given by

ψ(ω) = sin(πω

2Ω

)

e−(ω−Ω)2

2 .

The wavelet transform of the signal is computed as:

Wf (a, t) =∫

f (x)ψ(

x − t

a

)

dx

a,

where ψ is the complex conjugate of ψ, t ∈ R stands for the location/timeparameter and a > 0 denotes the scale parameter.



Empirical Wavelet Mode Decomposition (EWMD) – WIME

a) Perform the continuous wavelet transform Wf (a, t) of f .

b) Compute the wavelet spectrum Λ associated to f :

Λ(a) = E |Wf (a, .)|

where E denotes the mean over time. Then look for the scale a∗ at whichΛ reaches its global maximum.

c) Extract the component related to a∗:

c1 = R−1ψ |Wf (a∗, t)|cos(argWf (a∗, t))

where Rψ ≈ 1.25 is a reconstruction constant.

d) Repeat steps (a) to (d) with f − c1 instead of f .

e) Stop the process when Λ(a∗)< ε for a threshold ε or at your convenience.The components successively extracted reconstruct f accurately.


Reconstruction skills

Table of contents




4 Real-life data


6 Conclusion




First example:

1 1024−1.5

0

1.5 −1

0

1

−1

0

1

−1

0

1

−1

0

1

Sum of 2 triangular waveforms and a sine wave (classic example of the EMD).




time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c1

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c2

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c3

First row: |Wf (a, t)|, spectrum of f , first extracted component c1. Second andthird rows: the same for f − c1 and f − c1 − c2.




WIME Expected EMD

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

First column: components extracted with WIME. Second column: the real(expected) components. Third column: the IMFs extracted with the EMD. Thereconstruction c1 + c2 + c3 has a correlation of 0.992 with the original signaland a RMSE of 0.085.


Period detection skills

Table of contents




4 Real-life data


6 Conclusion



Period detection

We consider the AM-FM signal f (x) = ∑4i=1 fi(x) where

f1(x) =

(

1+0.5cos

(

2π200

x

))

cos

(

2π47

x

)

(1)

f2(x) =ln(x)

14cos

(

2π31

x

)

(2)

f3(x) =

√x

60cos

(

2π65

x

)

(3)

f4(x) =x

2000cos

(

2π23+ cos

(

2π1600 x

)x

)

. (4)

Target periods: ≈ 23, 31, 47, 65 units.



Period detection

1 200 400 600 800

−2

−1

0

1

2

−101

−101

−101

1 200 400 600 800−101

Target periods: ≈ 23, 31, 47, 65 units.



Period detection - WIME

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c1

time

freq

uenc

y

spectrumfr

eque

ncy

−1

0

1

time

c2

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c3

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c4

Target periods: ≈ 23, 31, 47, 65 units. Detected periods corresponding to a∗:46.4, 30.6, 65.5 and 21.6 units. Correlation: 0.996. RMSE: 0.069.



Period detection - EMD

−1

0

1

−1

0

1

1 200 400 600 800

−1

0

1

1 200 400 600 800

−1

0

1

Target periods: ≈ 23, 31, 47, 65 units. Periods extracted from theHilbert-Huang transform: 41, 75, 165, 284 units.


Real-life data

Table of contents




4 Real-life data


6 Conclusion


Real-life data

ENSO index

Analyzed data: Niño 3.4 time series, i.e. monthly-sampled sea surfacetemperature anomalies in the Equatorial Pacific Ocean from Jan 1950 toDec 2014 (http://www.cpc.ncep.noaa.gov/).


http://www.cpc.ncep.noaa.gov/

Real-life data

ENSO index

Niño 3.4 index:

1950 1960 1970 1980 1990 2000 2010 2020

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

time (years)

Nin

o 3.

4 in

dex

17 El Niño events: SST anomaly above +0.5C during 5 consecutivemonths.14 La Niña events: SST anomaly below −0.5C during 5 consecutivemonths.


Real-life data

ENSO index

Flooding in the West coast of South America

Droughts in Asia and Australia

Fish kills or shifts in locations and types of fish, having economic impactsin Peru and Chile

Impact on snowfalls and monsoons, drier/hotter/wetter/cooler than normalconditions

Impact on hurricanes/typhoons occurrences

Links with famines, increase in mosquito-borne diseases (malaria,dengue, ...), civil conflicts

In Los Angeles, increase in the number of some species of mosquitoes (in1997 notably).

...


Real-life data

ENSO index

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c1

time

freq

uenc

y

spectrumfr

eque

ncy

−1

0

1

time

c2

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c3

time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

time

c4time

freq

uenc

y

spectrum

freq

uenc

y

−1

0

1

timec5


Real-life data

ENSO index

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−101

−101

−101

−101

−101

Components extracted with WIME and the IMFs from the EMD. Periods: 44.8,28.6, 17, 65.6, 140.6 months (WIME) and 9.8, 21, 38.6, 75.9, 138.4 (EMD).Those from WIME are more in agreement with some previous works.


Recent improvements

Table of contents




4 Real-life data


6 Conclusion


Recent improvements

Recent improvements

Problems with “highly” non-stationary components (classic example of theEMD):

1 2000−0.5

0

0.5

1

−0.50

0.5

−0.50

0.5

−0.50

0.5

−0.50

0.5

−0.50

0.5

−0.50

0.5

Reconstruction is excellent (RMSE = 0.08, PCC = 0.97) but components arenot correct.


Recent improvements

Recent improvements

Solution: add flexibility, follow ridges of maxima.

0 0.5 1

204080

160320640

time

freq

uenc

y

0 0.2 0.41 0.62

204080

160320640

spectrum

freq

uenc

y

0 0.5 1

−1

0

1

time

c1

0 0.5 1

−1

0

1

time

f1

0 0.5 1

204080

160320640

time

freq

uenc

y

0 0.2 0.41 0.62

204080

160320640

spectrum

freq

uenc

y

0 0.5 1

−1

0

1

timec2

0 0.5 1

−1

0

1

time

f2

0 0.5 1

204080

160320640

time

freq

uenc

y

0 0.2 0.41 0.62

204080

160320640

spectrum

freq

uenc

y

0 0.5 1

−1

0

1

time

c3

0 0.5 1

−1

0

1

time

f3Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016

Recent improvements

Recent improvements

Crossings in the TF plane:

f1(t) = 1.25cos(

−(t −3)3 +180t)

(5)

f2(t) = cos(

1.8t +20t)

(6)

f3(t) =

0.75cos(−1.6πt2 +20πt) (t < 5)0.75cos(4πt) (t ≥ 5)

(7)

0 10 20−3−2−10123

time

f


Recent improvements

Recent improvements

WIME

0 10 20

1.252.5

510204080

time

freq

uenc

y

0 0.39 0.78 1.17

1.252.5

510204080

spectrum

freq

uenc

y

0 10 20−2

−1

0

1

2

time

c1

0 10 20−2

−1

0

1

2

time

f1

0 10 20

1.252.5

510204080

time

freq

uenc

y

0 0.39 0.78 1.17

1.252.5

510204080

spectrum

freq

uenc

y

0 10 20−2

−1

0

1

2

timec2

0 10 20−2

−1

0

1

2

time

f2

0 10 20

1.252.5

510204080

time

freq

uenc

y

0 0.39 0.78 1.17

1.252.5

510204080

spectrum

freq

uenc

y

0 10 20−2

−1

0

1

2

time

c3

0 10 20−2

−1

0

1

2

time

f3Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016

Recent improvements

Recent improvements

EMD

0 10 20

1.252.5

510204080

time

freq

uenc

y

0 10 20

−2

−1

0

1

2

time

IMF

1

0 10 20

1.252.5

510204080

timefr

eque

ncy

0 10 20

−2

−1

0

1

2

time

IMF

2

0 10 20

1.252.5

510204080

time

freq

uenc

y

0 10 20

−2

−1

0

1

2

time

IMF

3


Conclusion

Table of contents




4 Real-life data


6 Conclusion


Conclusion

Conclusion

WIME

is a wavelet-based decomposition method

has reconstruction skills comparable to the EMD

has period detection skills better than the EMD in mode-mixing problems

handles crossing patterns in the TF plane

gives accurate results with real-life signals

Future works: study the mathematical properties of the method, test itstolerance to noise, limit edge effects,...

Thanks for your attention.

Please refer to the paper for relevant references.


Documents

A New Wavelet-Based Mode Decomposition for …...A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition Adrien DELIÈGE